José Ángel Cid

Computation of Green’s functions for boundary value problems with Mathematica☆

Abstract This paper is devoted to construct an algorithm that allows us to calculate the explicit expression of the Green’s function related to a n th-order linear ordinary differential equation, with constant coefficients, coupled with two-point linear boundary conditions. We develop this algorithm by making a Mathematica package.

On the sign of the Green's function associated to Hill's equation with an indefinite potential

Abstract In this note we give a L p -criterium for the positiveness of the Green’s function of the periodic boundary value problem: x ″ + a ( t ) x = 0 , x ( 0 ) = x ( T ) , x ′ ( 0 ) = x ′ ( T ) with and indefinite potential a ( t ) . Moreover, we prove that such Green’s function is negative provided a ( t ) belongs to the image of a suitable periodic Ricatti type operator.

A positive fixed point theorem with applications to systems of Hammerstein integral equations

We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of positive solutions for systems of nonlinear Hammerstein integral equations. An example is also presented to show the applicability of our results.MSC: 47H10, 34B10, 34B18, 45G15, 47H30.

New criteria for the existence of non-trivial fixed points in cones

We give new criteria for the existence of nontrivial fixed points on cones assuming some monotonicity of the operator on a suitable conical shell. Moreover, we give an application to the existence of multiple solutions for a nonlocal boundary value problem that models the displacement of a beam subject to some feedback controllers.MSC:47H10, 34B10, 34B18.

Existence and Multiplicity of Solutions for a Periodic Hill's Equation with Parametric Dependence and Singularities

We deal with the existence and multiplicity of solutions for the periodic boundary value problem , where is a positive parameter. The function is allowed to be singular, and the related Greens function is nonnegative and can vanish at some points.

A TOPOLOGICAL APPROACH TO PERIODIC OSCILLATIONS RELATED TO THE LIEBAU PHENOMENON

Abstract We give some sufficient conditions for existence, non-existence and localization of positive solutions for a periodic boundary value problem related to the Liebau phenomenon. Our approach is of topological nature and relies on the Krasnoselskiĭ–Guo theorem on cone expansion and compression. Our results improve and complement earlier ones in the literature.

Does Lipschitz with Respect to x Imply Uniqueness for the Differential Equation y' = f(x,y)?

Of course, the answer to the question posed in the title is no, in general, but, surprisingly enough, yes in a significant situation to be detailed later. This paper aims to bring to the attention of the widest possible mathematical audience an elementary result, based on the theorem of differentiation of inverse functions, which extends the applicability of uniqueness theorems. Loosely speaking, it transforms every uniqueness theorem into an alternative version of it with the roles of the dependent and the independent variables interchanged in the assumptions. A remarkable example of such a theorem is the version of Lipschitz’s theorem alluded to in the first paragraph, but there are at least as many new possibilities as old uniqueness theorems.

New results for the Liebau phenomenon via fixed point index

Abstract We prove new results regarding the existence of positive solutions for a nonlinear periodic boundary value problem related to the Liebau phenomenon. As a consequence we obtain new sufficient conditions for the existence of a pump in a simple model. Our methodology relies on the use of classical fixed point index. Some examples are provided to illustrate our theory. We improve and complement previous results in the literature.

Green's functions and spectral theory for the Hill's equation

The aim of this paper is to show certain properties of the Greens functions related to the Hills equation coupled with various two point boundary value conditions. We will obtain the expression of the Greens function of Neumann, Dirichlet, Mixed and anti-periodic problems as a combination of the Greens function related to periodic ones.As a consequence we will prove suitable results in spectral theory and deduce some comparison results for the solutions of the Hills equation with various boundary value conditions.

A simple method to find out when an ordinary differential equation is separable

We present an alternative method to that of Scott (D. Scott, When is an ordinary differential equation separable?, Amer. Math. Monthly 92 (1985), pp. 422–423) to teach the students how to discover whether a differential equation is separable or not when the nonlinearity f(x, y) is not explicitly factorized. Our approach is completely elementary and provides a necessary and sufficient condition for separability, as well as simple formulas for the explicit factors of f(x, y).